Contact Stiffness Study: Modelling and Identification...stiffness has been widely accepted. With the consideration on the contact and friction conditions, the validity and accuracy

12

Contact Stiffness Study: Modelling and Identification

Hui Wang1, Yi Zheng2 and Yiming (Kevin) Rong1,2 1Tsinghua University,

2Worcester Polytechnic Institute, 1China

2USA

1. Introduction

In machining processes, fixtures are used to accurately position and constrain workpiece relative to the cutting tool. As an important aspect of tooling, fixturing significantly contributes to the quality, cost, and the cycle time of production. Fixturing accuracy and reliability is crucial to the success of machining operations.

Computerized fixture design &analysis has become means of providing solutions in production operation improvement. Although fixtures can be designed by using CAD functions, a lack of scientific tool and systematic approach for evaluating the design performance makes them rely on trial-and-errors, which leads to several problems, for instance, over design in functions, which is very common and sometimes depredates the performance (e.g., unnecessary heavy design); the quality of design that cannot be ensured before testing; the long cycle time of fixture design, fabrication, and testing, which may take weeks if not months; a lack of technical evaluation of fixture design in the production planning stage.

Over past two decades, Computerized Aided Fixture Design (CAFD) has been recognized as an important area and studied from fixture planning, fixture design to fixturing analysis/verification. The fixture planning is to determine the locating datum surfaces and locating/clamping positions on the workpiece surfaces for a totally constrained locating and reliable clamping. The fixture design is to generate a design of fixture structure as an assembly, according to different production requirements such as production volume and machining conditions. And the design verification is to evaluate fixture design performances for satisfying the production requirements, such as completeness of locating, tolerance stack-up, accessibility, fixturing stability, and the easiness of operation.

For many years, fixture planning has been the focus of fixture related academic research with significant progress in both theoretical and practical studies. Most analyses are based on strong assumptions, e.g., frictionless smooth surfaces in contact, rigid fixture body, and single objective function for optimization. Fixture design is a complex problem with considerations of many operational requirements. Four generations of CAFD techniques and systems have been developed: group technology (GT)-based part classification for

www.intechopen.com

Finite Element Analysis – From Biomedical Applications to Industrial Developments

320

fixture design and on-screen editing, automated modular fixture design, permanent fixture design with predefined fixture components types, and variation fixture design for part families. The study on a new generation of CAFD just started to consider operational requirements. Both geometric reasoning, knowledge-based as well as case-based reasoning (CBR) techniques have been intensively studied for CAFD. How to make use of the best practice knowledge in fixture design and verify the fixture design quality under different conditions has become a challenge in the fixture design &analysis study.

In fixture design verification, it was proved that when the fixture stiffness and machining force are known as input information, the fixturing stability problem could be completely solved. However most of the studies were focused on the fixtured workpiece model, i.e., how to configure positions of locators and clamps for an accurate and secured fixturing. FEA method has been extensively used to develop fixtured workpiece model (e.g., Fang, 2002; Lee, 1987; Trappey, 1995) with an assumption of rigid or linear elastic fixture stiffness. The models and computational results cannot represent the nonlinear deformation in fixture connections identified in previous experiments. As Beards (1983) pointed out, up to 60% of the deformation and 90% of the damping in a fabricated structure can arise from various connections. The determination of fixture contact stiffness is the key barrier in the analysis of fixture stiffness. The existing work is very preliminary, by either simply applying the Hertzian contact model or considering the effective contact area.

The development of fixture design &analysis tools would enhance both the flexibility and the performance of the workholding systems by providing a more systematic and analytic approach to fixture design. Fixture stationary elements, such as locating pads, buttons, and pins, immediately contact with the workpiece when loading the workpiece. Subsequent clamping (by moveable elements) creates pre-loaded joints between the workpiece and each fixture component. Besides, there may be supporting components and a fixture base in a fixture. In fixture design, a thoughtful, economic fixture-workpiece system maintains uniform maximum joint stiffness throughout machining while also providing the fewest fixture components, open workpiece cutting access, and shortest setup and unloading cycles. Both static and dynamic stiffness in this fixture-workpiece system rely upon the component number, layout and static stiffness of the fixture structure. These affect fixture performance and must be addressed through appropriate design solutions integrating the fixture with other process elements to produce a highly rigid system. This requires a fundamental understanding of the fixture stiffness in order to develop an accurate model of the fixture - workpiece system.

2. Computer-aided fixture design with predictable fixture stiffness

The research on fixture-workpeice stiffness is a crucial topic in fixture design field. Currently, based on the elastic body assumption, using FEA method to predict the fixture stiffness has been widely accepted. With the consideration on the contact and friction conditions, the validity and accuracy of the methodology was been illustrated by two cases simulation and experimental comparison (Zhu, 1993).

The following is an introduction on the general methodology.

First the stiffness of typical fixture units is studied with considerations of contact friction conditions. The results of the fixture unit stiffness analysis are integrated in fixture design as

www.intechopen.com


321

a database with variation capability driven by parametric representations of fixture units. When a fixture is designed using fixture design &analysis tool, the fixture stiffness at the contact locations (locating and clamping positions) to the workpiece can be estimated and/or designed based on the machining operation constraints (e.g., fixture deformation and dynamic constraints). Fig. 1 shows a diagram of the integrated fixture design system.

Fig. 1. Integrated Fixture Design System

In order to study the fixture stiffness in a general manner, fixture structure is decomposed into functional units with fixture components and functional surfaces (Rong, 1999). In a

www.intechopen.com


322

fixture unit, all components are connected one to another where only one is in contact directly with fixture base and one or more in contact with the workpiece serving as the locator, clamp, or support. Fig. 2 shows a sketch of the fixture units in a fixture design. When a workpiece was located and clamped in the fixture, the fixture units are subjected to the external loads that pass through the workpiece. If the external load is known and acting on a fixture unit, and the displacement of the fixture unit at the contact position is measured or calculated based on a finite element (FE) model, the fixture unit stiffness can be determined.

The fixture unit stiffness is defined as the force required for a unit deformation of the fixture unit in normal and tangential directions at the contact position with workpiece. The stiffness can be static if the external load is static (such as clamping force), and dynamic if the external load is dynamic (such as machining force). It is the key parameter to analyze the relative performance of different fixture designs and optimize the fixture configuration.

Analysis of fixture unit stiffness may be divided into three categories: analytical, experimental and finite element analysis (FEA). Conventional structural analysis methods may not work well in estimating the fixture unit stiffness. Preliminary experimental study has shown the nature of fixture deformation in T-slot based modular fixtures (Zhu, 1993). An integrated model of a fixture-workpiece system was established for surface quality prediction (Liao, 2001) based on the experiment results in (Zhu, 1993), but combining zhu’s experimental work and finite element analysis (FEA). Hurtado used one torsional spring, and two linear springs, one in the normal direction and the other in the tangential direction, to model the stiffness of the workpiece, contact and fixture element. (Hurtado, 2002) FEA method has not been studied for fixture unit stiffness due to the complexity of the contact conditions and the large computation effort for many fixture components involved.

Fig. 2. Sketch of Fixture Units

www.intechopen.com


323

3. Finite element model with frictional contact conditions

3.1 Finite element formulation

Consider a general fixture unit with two components I and J, as shown in Fig.3 (Zheng, 2005,

2008). For multi-component fixture units, the model can be expanded. The fixture unit is

discretized into finite element models using a standard procedure, except for the contact

surfaces, where each nodes on the finite element mesh for the contact surface is modelled by

a pair of nodes at the same location belonging to components I and J, respectively, which are

connected by a set of contact elements. The basic assumptions include that material is

homogenous and linearly elastic, displacements and strains are small in both components I

and J, and the frictional force acting on the contact surface follows the Coulomb law of

friction.

The total potential energy p of a structural element is expressed as the sum of the internal

strain energy U and the potential energy Ω of the nodal force; that is,

p U (1)

It is well known that the element strain energy can be expressed as,

1

2

TU q K q (2)

where K is the element stiffness matrix; and q is the element nodal displacement vector.

The potential energy of the nodal force is,

Tq R (3)

where R is the vector of the nodal force. It includes internal force and external force.

When the two components I and J are in contact, a number of three-dimensional contact elements are in effect on the contact surfaces. It should note that the problem is strongly nonlinear, partially due to the fact that the number of contact elements may vary with the change of contact condition. The original contacting nodes might separate or recontact after separation, based on the deformation condition on the contact surface; also contact stiffness may not constant either. The contact elements are capable of supporting a compressive load in the normal direction and tangential forces in the tangential directions. When the two components are in contact, and the displacements in the tangential directions and normal direction are assumed as independent, the element itself can be treated as three independent contact springs: two having stiffness kt and k in the tangential directions of the contact surface at the contact point and one having stiffness kn in the normal direction.

Usually, there are two methods used to include the contact condition in the energy equation:

the Lagrange multiplier and the penalty function methods. In order to understand these

methods, a physical model of the contact conditions is presented, shown in Fig. 4. When two

contact surfaces of fixture components, i.e., body J and I, are loaded together, they will

contact at a few asperities.

www.intechopen.com


324

Fig. 3. Contact Model of Two Fixture Components

The contact criteria can be written as:

0; 0; 0ni nif f

Where, is distance from a contact point i in body I to a contact point j on the body J in the normal direction of contact; fni is the contact force acting on point i of body I in the normal direction.

www.intechopen.com


325

It shows the kinematic condition of no penetration and the static condition of compressive

normal force. To prevent interpenetration, the separation distance for each contact pair

must be greater or equal to zero. If >0, the contact force fni=0. When =0, the points are in

contact and fni<0. If <0, penetration occurs. In real physics, the actual contact area increases,

and contact stiffness is enhanced when the load increases. Therefore, the contact

deformation is nonlinear as a function of the preload as shown Figure 4(e). In the Lagrange

multiplier method, the function w(, fni) represents the constraint, which prevents the

penetration between contact pairs. In the penalty function method, an artificial penalty

parameter is used to prevent the penetration between contact pairs.

Body I

Body JR

E1'v1

E2'v2

a) b)

η

R

Initial State c)

State after loading d)

Normal contact deformation curve e)

Fig. 4. Physical Model of the Contact Conditions

In the penalty function method, the contact condition is represented by the constraint equation,

www.intechopen.com


326

Ct K q Q (4)

Where {t} is the constraint equation, CK is the contact element stiffness matrix, Q is the

contact force vector of the active contact node pairs. When 0t , it means that the

constraints are satisfied. So the constraint equation Eq. 4 becomes

CK q Q (5)

The total potential energy Πp in Eq. 1 can be augmented by a penalty function 1

2

Tt t

where is a diagonal matrix of penalty value i . The total potential energy in the

penalty function method becomes

1 1Π2 2

T T TpP q K q q R t t (6)

The minimization of ΠpP with respect to q requires that Π0pP

q , which leads to

T TC C CK K K q R K Q (7)

where TC CK K is the penalty matrix.

On the other hand, in the Lagrange multiplier method, the contact constraint equation can be written as:

TCw K q Q (8)

where the components of the row vector i (i=1, 2, …, N), are often defined as Lagrange

multipliers i .

Adding Eq. 8 to the potential energy in Eq. 1, we have the total energy in the Lagrange multiplier method,

1Π2

T T TpL Cq K q q R K q Q (9)

The minimization of ΠpL with respect to q and requires that Π0pL

q and

Π0pL , which leads to,

Π0

TpLCK q K R

q (10)

www.intechopen.com


327

Π0pL

CK q Q (11)

In a matrix form, Eqs. 10 and 11 can be expressed as,

0

TC

C

RqK K

QK (12)

While the constraints in Eq. 8 can be satisfied, the Lagrange multiplier method has disadvantages. Because the stiffness matrix in Eq. 12 may contain a zero component in its diagonal, there is no guarantee of the absence of the saddle point. In this situation, the computational stability problem may occur. In order to overcome that difficulty, a perturbed Lagrange multiplier method was introduced (Aliabadi, 1993).

1Π Π

21 1

2 2

TppL pL

T T T TCq K q q R K q Q

(13)

where is an arbitrary positive number. At the limit goes to , the perturbed solutions

converge to the original solutions. The introduction of will maintain a small force across

and along the interface. This will not only maintain stability but also avoid the stiffness

matrix being singular, due to rigid body motion. Similarly, the minimization of ppL with

respect to q and results in the following matrix,

1

TC

C

K K Rq

QK I

(14)

Eq. 14 can be expressed as:

TCK q R K (15)

CK q Q (16)

Substitute Eq. 16 into Eq. 15,

T TC C CK K K q R K Q

For simplicity, let all i in [] of penalty function equal to , i.e. i = . Thus, the

perturbed Lagrange multiplier is equivalent to the penalty function method.

In the Lagrange multiplier method, both displacement and contact force are regarded as independent variables; thus, the constraint (contact) conditions can be satisfied and the contact force can be calculated. It has disadvantages. The stiffness matrix contains zero

www.intechopen.com


328

components in its diagonal, and the Lagrange multiplier terms must be treated as additional variables. This leads to the construction of an augmented stiffness matrix, the order of which may significantly exceed the size of the original problem in the absence of constraint equations (Aliabadi, 1993). In comparison with the Lagrange multipliers method, the implementation of the penalty function method is relatively simple and does not require additional independent variables. It is often adopted in the practical analysis because of its simple implementation.

3.2 Contact conditions

Based on an iterative scheme (Mazurkiewicz, 1983), the contact conditions in FEA model are classified into the following three cases:

1. Open condition: gap remains open; 2. Stick condition: gap remains closed, and no sliding motion occurs in the tangential

directions; and 3. Sliding condition: gap remains closed, and the sliding occurs in the tangential

directions.

Let fji and uji be the contact nodal load vector and the nodal displacement, respectively, which are defined in the local coordinate system, where the subscript j indicates the component number ( j = I or J), and i indicates the coordinate (i = n, t, τ), as shown in Fig. 5.

By equilibrium of the contact element, 0In It I Jn Jt Jf f f f f f . Fi (i = n, t, τ) is the

external nodal load in i direction 1

nn

tx

x

F

R F

F

where x is the node number of body I or J.

The displacement and force must satisfy the equilibrium equations in the three contact conditions (note that {n, t, τ} is the local coordinate system).

a) b)

Fig. 5. Sketch of Contact Force on the Contact Surface

www.intechopen.com


329

3.2.1 Open condition

When the normal nodal force Fn is positive (tension), the contact is broken, and no force is

transmitted. The displacement change in the normal and tangential directions, denoted

respectively by , ,iu i n t , then is

, 0 , ,n Jn In n Ji Iiu u u f f i n t (17)

where uJn and uIn are the current displacements of node J and node I in a normal direction,

respectively. For each structural contact element, stiffness and forces are updated, based

upon current displacement values, in order to predict new displacements and contact forces.

n is the gap between a pair of the potential contact points. In each increment of load, the

gap status and the stiffness values are iteratively changed until convergence. As the load is

increased, n will change and hence should be adjusted as 0 Tn n n , where 0

n is the

initial gap before any deformation and Tn is the gap change caused by the total combined

normal movement at the pair of points.

3.2.2 Stick condition

The force in the tangential direction SF , which is the composition of the nodal force in t

and directions (Ft and F), is defined only when 0nF (compression). When the absolute

value of SF is less than | |nF , where is the Coulomb friction coefficient, there is no

slide-motion in the interface, and the contact element responds like a spring. The stick

condition exists if | |n Jt t J It t IF u k u k u k u k . That is,

, 0 , 0 ,Ii Ji Jn In n Ji Iif f u u u u i t (18)

where kt and k are the tangential contact stiffness in t and directions, respectively. In the

analysis of fixture unite stiffness, set tk k .

3.2.3 Sliding condition

Slide-motion will occur when the absolute value of SF is more than | |nF . The slide-

motion may occur in both the element t and directions. That is, if | |n Jt t J It t IF u k u k u k u k , then,

, , , 0It Jt n I J n In Jn In Jn ntf f F f f F f f u u (19)

where n tF and nF mean the maximum friction force in t and directions.

3.3 Solution procedure

The model presented in the previous section can be implemented to determine the fixture unit stiffness in clamping and machining. Because the model involves high nonlinearity, the

www.intechopen.com


330

Newton-Raphson (N-R) approach is used to solve the problem. Considering the full Newton-Raphson iteration it is recognized that in general the major computational cost per iteration lies in the calculation and factorization of the stiffness matrix. Since these calculations can be quite expensive when large-order systems are considered, the modified Newton-Raphson algorithm is used in this research (Bathe, 1996).

Given the applied load R and the corresponding displacement u, the applied load is divided into a series of load increments. At each load step, the contact stiffness and contact conditions remain constant. And several iterations may be necessary to find a solution with acceptable accuracy. The modified Newton-Raphson method is used first to evaluate the initial out-of-balance load vector at the beginning of the iteration at each load step. The out-of-balance load vector is defined as the difference between the applied load vector R and the

vector of restoring loads riR . When the out-of-balance load is non-zero, the program

performs a linear solution, using the initial out-of-balance loads, and then checks for convergence. If the convergence criteria are not satisfied, the out-of-balance load vector is reevaluated, the new contact conditions and the stiffness matrix are updated, and a new solution is obtained. This iterative procedure continues until the solution converges. The modified Newton-Raphson method and its flowchart are outlined by Fig.6.

Fig. 6. (a) Modified Newton-Raphson Method

www.intechopen.com


331

Input:

Fixture unit CAD model

Material properties

External forces

Finite element model:

Define Element Type and FEA Mesh

Apply Boundary Condition

Set Initial Contact Condition

Load Increment

Identify the element stiffness

matrix

Calculate the displacements of all

substructure

Check if contact

No

Yes

Update

Displacement,

Reaction force,

Contact force.

Final Load Step?

Output:

Global stress and displacement

Displacement of contact surfaces

Reaction forces

Contact forces

Yes

Identify penalty terms

Calculate contact force

Status Converged?Yes

No

No

Redefine Contact Element

stiffness

Fig. 6. (b) Flow Chart of the Analysis Procedure

4. Contact stiffness identification using a dynamic approach

First, the dynamic method is studied for use in the estimation of normal contact stiffness. The results of the dynamic methods are compared with the results based on the static test of normal contact stiffness; then the validated dynamic test method is used in estimation of tangential contact stiffness.

4.1 Theoretical formulation of 1-D normal contact stiffness

The idea behind the identification of normal contact stiffness is that the contact interface is modeled by a discrete linear spring. When the preload is changed, contact stiffness will

www.intechopen.com


332

change. When body I is in contact with the ground, the dynamic model of the entire structure can be shown as in Fig.7. According to this theoretical model the relationship between natural frequencies and normal contact stiffness can be established. When natural frequencies are obtained from impact test, along with a theoretical model, normal contact stiffness can be estimated.

In the one-dimensional model of body I, m is the mass of body I, k is the contact stiffness, p is the preload, f(t) is impulse excitation, u(x,t) is the longitudinal displacement of the bar at distance x from a fixed reference.

Fig. 7. One-Dimensional Model for Normal contact Stiffness

With use of a bar in Fig.7, the governing equation of the longitudinal vibration of the bar can be expressed as

2

2

, ,u x t u x tA EA

x xt (20)

The boundary conditions of the bar are:

At x=0: 0,0

u tEA

x

(21)

and at x=l: ,

n

u l tEA k u

x

(22)

Initially, the system starts from rest, from the static equilibrium position of the bar, such that the initial displacement condition is:

At t=0, ,0 0u x (23)

The response of a system to an impulsive force can also be obtained by considering that the impulse produces an instantaneous change in the momentum of the system before any appreciable displacement occurs. The second initial condition is

,0 1u x

t m

(24)

Assume ,u x t X x q t (25)

www.intechopen.com


333

Substitute Eq. 25 into Eq. 20 to obtain

2 22

2 2

2 22

2 2 2

1 1

d q t d X xX x c q t

dt dx

d X x d q t

X x dx c q t dt

(26)

-2 is called the separation constant and is designated to be negative (De Silva, 1999).

Therefore, the mode shapes X (x) satisfies

2

22

0d X x

X xdx

(27)

whose general solution is 1 2sin cosX x C x C x (28)

According to the general solution and the modal boundary conditions, one can get

tan nkl

EA (29)

Set the structure stiffness as * EAk

l and the ratio of the stiffness as

*nk

k . Since the

structure stiffness k* is constant and known, the ratio of the stiffness is proportional to the contact stiffness kn. Therefore Eq. 11 can be expressed as

*

tan

nk

kll l

(30)

This transcendental equation has an infinite number of solutions i (i=1,2,…)that correspond to the modes of vibration. When is changed, the solution of i will change. When is changed from 0.1 to 10, one can get the corresponding il as shown in Fig.8. The natural frequencies can also be obtained using

i i iE

c (31)

In an experimental study, the natural frequencies can be obtained by an impact test. i can calculated from Eq.31 since the natural frequencies are related to the system characteristics.

Then can be determined from Eq. 30. Finally, the contact stiffness, kn, can be estimated

based on the definition of . According to the assumption that contact stiffness is a function of the preload, the natural frequencies can be determined in experiments under different preloads. The change of contact stiffness can then be identified based on the change of the preloads, through measurement of the natural frequency variation. It should be noted that although any mode of the natural frequency can be used to estimate the contact stiffness, some modes might be more sensitive than others to the change of the preloads.

www.intechopen.com


334

(a) First Mode

(b) Second Mode

(c) Third Mode

(d) Fourth Mode

Fig. 8. Relationships between the Nondimensional Natural Frequencies and the Stiffness

Ratio in the First Four Modes

4.2 Experimental procedure and results

The experiments were conducted in order to verify the method of identifying contact

stiffness in the normal direction (Zheng, 2005, 2008). The measurement instrumentation

includes the proximity, the impact hammer with a load cell, power supply, and a Fast

Fourier transformation (FFT) analyzer, as shown in Fig.9. The experimental procedure can

be expressed as follows:

1. Frequency response function (FRF) of the bar is measured by using the hammer to excite the system. Thus, the natural frequencies of the bar can be obtained.

2. According to the natural frequency equation i iE , i is calculated.

3. Based on the relationship between il and of the first three modes in Fig.8, the can be inferred from the comparison of experimental results and theoretical results. Then the

normal contact stiffness can be obtained from the equation *nk

k .

When the natural frequencies are obtained from the experiment, along with the curves of

the relationships between il and , contact stiffness can be determined from each mode of

vibration. However, when the preload changes, the natural frequencies may not necessarily

change significantly with the change of normal load for certain modes. Contact stiffness

should be identified from the mode most sensitive to changes of a preload. Fig.10 shows the

FRF of the test system under different preload. Fig.11 shows the relationships between the

www.intechopen.com


335

natural frequencies and preload. The natural frequency of the third mode f3 is the most sensitive to changes in a preload.

Fig. 9. Measurement Setup

Fig. 10. A simplifed illustration on the Frequency Response Functions (FRF) of the Test System

www.intechopen.com


336

Fig. 11. The relationships between the natural frequencies and preloads

Once the natural frequency is obtained from the test, contact stiffness can be estimated by calculating il, , and kn. In order to verify the results, these calculations were compared with the previous static measurement results of contact stiffness. Under the same experimental condition, i.e., the same experimental device and preloads, contact stiffness is obtained and used in the calculation of natural frequencies, and then compared with the results of dynamic tests, as shown in Fig.12.

(a) Natural Frequency vs. Preloads (b) Normal Contact Stiffness vs. Preloads

Fig. 12. Comparisons between Experimental Result of Dynamic Test and Numerical Value Based on Static Test

www.intechopen.com


337

It can be seen that the results from the dynamic tests are consistent with the numerical calculation results based on the static test results. When the results of the dynamic test are consistent with the static test results, the dynamic test method can be used in identifying tangential contact stiffness, for which the static tests are too difficult to conduct.

4.3 Theoretical formulation of tangential contact stiffness

Two fixture components are in contact at a certain number of asperities due to the inherent roughness of the surface. When they are subjected to tangential forces, the components are mutually constrained through frictional contacts. A friction model based on the Coulomb friction theory is shown in Fig.13. The tangential contact stiffness results from the elasticity of asperities of the contact surfaces, and the total resulting stiffness of these contact surfaces depends on their statistical topographical parameters.

uP

-uP

Slide area Stick area Slide area

Fig. 13. A Friction Model

Consider that body I is brought into contact with the flat surface of the support under a uniform preload, P, and is subjected to an small excitation, F, as shown in Figure 14. It is assumed that the tangential contact stiffness will change as the preload increases. The friction at each contact point is governed by Coulomb’s law. When force is applied in the tangential direction, the asperities in body I will also deform until the shear stress between the asperities exceeds the limit, then the contact surface will slide each other. The friction model of body I in contact is shown in the Fig.15. The friction force is given by Eq. 32.

/t tK u u P K

fP otherwise

(32)

The idea of the identification of the tangential contact stiffness is to compare the two sets of system natural frequencies: one set is identified from the measured impulse response in tangential direction under different preloads, and the other set is calculated from the FEA model of the system. Based on the numerical simulation, a relationship between tangential contact stiffness and the natural frequencies can be established. If the natural frequencies are measured in the experiments under different preloads, the contact stiffness can be calculated from the relationship obtained by the numerical simulation.

www.intechopen.com


338

Fig. 14. Body I on the Support

Fig. 15. Friction Model of the Fixture Components in Contact

In order to do the numerical simulation, the effect of the contact force needs to be included into FEA model of the system. The additional contact stiffness matrix will be introduced in the general FEA model. The derivation of contact stiffness matrix is briefly given as follows.

Consider an elastic body I in Fig.15, the kinetic, strain, and potential energies of the system can be expressed respectively as:

2

T

V

u uK dV

t t

(33)

1

2

T

V

U dV (34)

1

ˆ ˆ

C

T T

c

S S

W F u dS R u dS (35)

www.intechopen.com


339

where K is the kinetic energy; {u} is the displacement vector; V is the volume of the elastic

body I; is the mass density of the material; U is the strain energy; and {}; are the strain

and stress components, respectively. W is the potential energy of external forces; F̂ is the

external surface force vector specified on the boundary S1; ˆcR is the contact force vector

on the contact surface Sc. Note that 1cS S ; The body force is ignored. Using the

above energy expressions the total potential energy of the system is

Π K W U (36)

Based on the well-known Hamilton’s principle, a discretized FEA formulation for a typical element can be expressed as

0e e eCM d t K K d t F (37)

To obtain the matrix form, the displacement field of a typical element {u}, which is a function of both space and time, can be written as:

u N x d t u N x d t u N x d t (38)

where [N(x)] is a vector of the space function; and {d(t)} is the nodal response vector. Using

the interpolation relationship the element,

Mass matrix is given by

e

Te

V

M N x N x dV (39)

And the element stiffness matrix is

e

Tee

V

K B D B dV (40)

where [B] is the geometry matrix.

Comparing to the standard FEA formulation an additional term of CK , referred as the

contact stiffness matrix in included in Eq. (37). The term stems from the work done by the

contact force on the contact surface. A brief derivation is presented as follows.

The work done by the contact force on the contact surface can be written as

ˆ

Ce

Tce ce

S

E u R dS (41)

Using the contact element, the contact force can be expressed as

ˆce cR D u (42)

Substituting Eqs. (38) and (42) into (41) yields

www.intechopen.com


340

Ce

TTce c

S

E d t N x D N x dS d t (43)

Therefore, the contact stiffness matrix can thus be defined as

Ce

TC c

S

K N x D N x dS (44)

where [Dc] is the contact property matrix. In the section, the displacements of contact

element in the normal direction are assumed to keep stick. Therefore, the normal contact

stiffness becomes infinity. The tangential contact stiffness is considered.

The derived contact stiffness matrix should be added to the general FEA model for the

fixture stiffness analysis to take into account the effects of the contact force. Followed the

standard procedure of the eigenvalue problem, the system natural frequencies can be

obtained using the FEA method to establish the relationship between the tangential contact

stiffness and natural frequencies. For example, a specimen that has the dimensions 530.75

in was used to measure dynamic characteristics. Fig.16 shows the FEA model of the

specimen. Contact elements were modeled as separate springs on the top and bottom

surfaces of the specimen. There are two nodes for each contact element. One node is on the

contact surface of the specimen. The other node is constrained at all degrees of freedom. The

impulse force was applied at the side of the specimen. The response was obtained at point

M, at the other side of the specimen.

Fig. 16. Finite Element Model of Specimen

Fig.17 shows the relationships between tangential contact stiffness and natural frequencies

of the first two vibration modes. The results are obtained through numerical simulation.

From experiments, the frequency response is measured under the different preloads. The

contact stiffness can be determined based on the relationships shown in Fig.17.

www.intechopen.com


341

Fig. 17. The Relationship of Tangential Stiffness vs. the First Two Natural Frequencies

5. Conclusions

Forces in a workpiece-fixture system have a crucial impact on the deformation and accuracy of the system. In this chapter, an FEA model of fixture unit stiffness is proposed. A contact model between fixture components are utilized for solving the contact problem encountered in the study of fixture unit stiffness. By several simple experiments and comparison with the corresponding analytical solution and experimental results in the literature, this methodology is validated. This analytic approach also can be extended in the research of complex fixture system with multiple units and components, which will lead to a new progress in the design and verification of fixture-workpiece system study.

6. References

Aliabadi, Brebbia C.A., 1993, Computational methods in contact mechanics, Computational Mechanics Publications.

Anthony C. Fischer-Cripps, Introduction to contact mechanics, Springer-Verlag Inc, New York. 2000.

Bathe K.J., 1996, Finite Element Procedures, Prentic-Hall, Inc. Beards C.F., 1983, The Damping of Structural Vibration by Controlled Inter-facial Slip in

Joints, Journal of Vibration, Acoustics, Stress, and Reliability in Design, 105, pp. 369–373 Fang B.; R.E. DeVor; S.G. Kapoor, 2002, Influence of friction damping on workpiece-fixture

system dynamics and machining stability, Journal of Manufacturing Science and Engineering, 124, pp. 226-233

Hurtado J. H.; S. N. Meltke, 2002, Modeling and Analysis of the Effect of Fixture-Workpiece Conformability on Static Stability, Journal of Manufacturing Science and Engineering, 124, pp. 234–241

www.intechopen.com


342

Lee, J. D.; Haynes L. S., 1987, Finite element analysis of flexible fixturing systems, Journal of Engineering for Industry, 109, pp.134-139

Liao, Y. G.; Hu S.J., 2001, An integrated model of a fixture-workpiece system for surface quality prediction, International Journal of Advanced Manufacturing Technology, 17, pp. 810-818

Mazurkiewicz M; W. Ostachowicz, 1983, Theory of finite element method for elastic contact problems of solid bodies, Computer Structure, Vol. 17, pp.51-59

Rong, Y.; Zhu Y., 1999, Computer-Aided Fixture Design, New York, NY, Dekker. Trappey A. J. C.; Su C. S., J. L. Hou, 1995, Computer-aided fixture analysis using finite

element analysis and mathematical optimization modeling, ASME, Manufacturing Engineering Division, MED, 2-1, pp.777-787

Zhu, Y.; S. Zhang, Y. Rong, 1993, Experimental study on fixturing stiffness of T-slot based modular fixtures, NAMRI Transactions XXI, NAMRC, Stillwater, OK, USA, pp. 231-235

Zheng Y., 2005, Finite Element Analysis for Fixture Stiffness, PhD Desertation, Worcester Polytechnic Institute, MA, USA.

Zheng Y.; Hou Z.; Rong Y., 2008, The study of fixture stiffness part I: a finite element analysis for stiffness of fixture units, International Journal of Advanced Manufacturing Technology, 36(9-10), pp.865-876

Zheng Y., Hou Z., Rong Y., 2008, The study of fixture stiffness -Part II: contact stiffness identification between fixture components, International Journal of Advanced Manufacturing Technology, 38(1-2), pp.19-31

www.intechopen.com

Finite Element Analysis - From Biomedical Applications toIndustrial DevelopmentsEdited by Dr. David Moratal

ISBN 978-953-51-0474-2Hard cover, 496 pagesPublisher InTechPublished online 30, March, 2012Published in print edition March, 2012

InTech EuropeUniversity Campus STeP Ri Slavka Krautzeka 83/A 51000 Rijeka, Croatia Phone: +385 (51) 770 447 Fax: +385 (51) 686 166www.intechopen.com

InTech ChinaUnit 405, Office Block, Hotel Equatorial Shanghai No.65, Yan An Road (West), Shanghai, 200040, China

Phone: +86-21-62489820 Fax: +86-21-62489821

Finite Element Analysis represents a numerical technique for finding approximate solutions to partialdifferential equations as well as integral equations, permitting the numerical analysis of complex structuresbased on their material properties. This book presents 20 different chapters in the application of FiniteElements, ranging from Biomedical Engineering to Manufacturing Industry and Industrial Developments. It hasbeen written at a level suitable for use in a graduate course on applications of finite element modelling andanalysis (mechanical, civil and biomedical engineering studies, for instance), without excluding its use byresearchers or professional engineers interested in the field, seeking to gain a deeper understandingconcerning Finite Element Analysis.

How to referenceIn order to correctly reference this scholarly work, feel free to copy and paste the following:

Hui Wang, Yi Zheng and Yiming (Kevin) Rong (2012). Contact Stiffness Study: Modelling and Identification,Finite Element Analysis - From Biomedical Applications to Industrial Developments, Dr. David Moratal (Ed.),ISBN: 978-953-51-0474-2, InTech, Available from: http://www.intechopen.com/books/finite-element-analysis-from-biomedical-applications-to-industrial-developments/contact-stiffness-study-modeling-and-identification

© 2012 The Author(s). Licensee IntechOpen. This is an open access articledistributed under the terms of the Creative Commons Attribution 3.0License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

http://creativecommons.org/licenses/by/3.0

Documents

Contact Stiffness Study: Modelling and Identification...stiffness has been widely accepted. With the consideration on the contact and friction conditions, the validity and accuracy